Agency incentives and. in regulating market risk. and. Simone Varotto

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1 Agency incentives and reputational distortions: a comparison of the eectiveness of Value-at-Risk and Pre-commitment in regulating market risk Arupratan Daripa and Simone Varotto * Birkbeck College, Department of Economics, 7-15, Gresse Street, London, W1P 2LL. adaripa@econ.bbk.ac.uk ** Bank of England, Threadneedle Street, London, EC2R 8AH. s.varotto@bankofengland.co.uk The views expressed are those of the authors and do not necessarily reect those of the Bank of England. The content of the paper as well as the exposition have beneted enormously from regular interactions with Patricia Jackson and Ian Michael. We have also beneted from comments on an earlier version by William Perraudin and two referees for the Bank of England Working Paper series. We are grateful to all of them. Issued by the Bank of England, Threadneedle Street, London, EC2R 8AH to which requests for individual copies should be addressed: envelopes should be marked for the attention of Publications Group. (Telephone: ). Bank of England 1997 ISSN

2 Contents Abstract 1 1 Introduction The background Hard links and soft links: a potential trade-o Separation of ownership and control in large banks: the agency problem Disclosure of breaches: reputational concerns 6 2 A summary of the results Agency incentives under a hard-link approach Agency incentives under PCA Loss aversion under reputational concerns and distortions in the choice of trading book size Modifying PCA: optimal regulation 8 3 The model The opportunity set Trading losses and the extent of liability Cost of regulatory capital 12 4 Agency incentives under hard-link approaches The bank's problem Regulatory control under a hard-link regime 14 5 Agency incentives under the pre-commitment approach (PCA) Agency incentives under PCA: the special case of q = Agency incentives under PCA: the general case 18 6 Agency distortion under PCA 19 7 Public disclosure of breach: reputational distortion under PCA An illustration 21 8 Conclusion: modifying PCA Modifying PCA to rectify the distortions 24

3 Appendix A 26 A.1 Regulatory control under a hard-link regime 26 A.2 Deriving equation (5.2) 27 A.3 The principal's payo in the special case 27 A.4 Proof of proposition 2 28 Appendix B 29 Removing assumption 1 29 References 32

4 Abstract In regulating the market risk exposure of banks, the approach taken to date is to use a `hard-link' regime that sets a relation between exposure and capital requirement exogenously. A new `pre-commitment' approach (PCA) proposes the use of a `soft-link'. Such a link is not externally imposed, but arises endogenously. Such an approach isofmuch greater economic appeal, as it is incentive-based and so less prescriptive. But, we argue that there is a trade-o. The use of incentives by the new approach implies that a whole host of strategic interactions in the bank are relevant in evaluating its eectiveness. This aspect of a soft-link regulation such as PCA seems to have received little attention. We attempt to clarify the precise nature of the trade-o by analysing two potential sources of distortion: agency and reputational. In the context of a simple principal-agent model, we study the incentives generated by PCA on managerial risk-taking when the level of risk is not directly observable by the bank owner. We identify contexts in which a distortion might arise. Second, we study the eect of reputational concerns under public disclosure of a breach. We show that this might lead to a perverse pattern in the relative size of the trading activities compared with the size of bank as a whole. A hard-link approach avoids such distortions. The results form a rst step towards modifying PCA to construct optimal incentive-compatible regulatory schemes. We discuss informally how PCA might be modied to rectify the distortions identied here.

5 1 Introduction 1.1 The background Traditionally, regulation of banks has focused on the risk carried by bank loans. Loans are typically non-traded assets. In recent years, another component of bank assets has become increasingly important: assets actively traded in the nancial markets. (1) These assets form the `trading book' of a bank, as distinct from the `banking book' that includes the non-traded assets such as loans. Though for most large banks the trading book is still relatively small compared with the banking book, its rising importance makes market risk of banks an important regulatory concern. Further, there is now amovetowards securitising even the traditional banking book assets, which adds to the importance of regulating the risk exposure of traded assets. In January 1996, the European Union adopted rules to regulate the market risk exposure of banks, setting risk-based capital requirements for the trading books of banks and securities houses. The approach taken is to use a `hard-link' regime that sets a relation between exposure and capital requirement exogenously. The adopted requirements, known as the standardised approach, laid down rules for calculating the capital requirement for each separate risk category (ie UK equities, US equities, UK interest rate risk and so on). These are added together to give the overall requirement. A weakness in this method is that it does not take into account the diversication benets of holding dierent risks in the same portfolio, and so yields an excessive capital requirement. One way to correct for this is to use the value-at-risk (VaR) models that some banks have developed to measure overall portfolio risk. The Basle Supervisors' Committee has now agreed to oer an alternative regime, with capital requirements based on such internal VaR models. Though the way of measuring risk employed by the two regimes is dierent, in both approaches the regulator lays down the parameters for the calculation of the capital requirement for a given exposure. So both embody a hard link. Under VaR, the capital requirement for a particular portfolio is calculated using the internal risk management models of the banks. (2) For any portfolio, the (1) For example, securities and foreign exchange or commodities positions that are held for short-term trading purposes. (2) The value-at-risk of a given portfolio can be calculated via parametric or non-parametric 2

6 aim is to estimate a level of potential loss over a particular time period which would only be exceeded with a given probability. The probability and the period are both laid down by the regulator. Basle has set these at 1% and 10 days. The capital requirement is based on this potential loss. Under Basle there is an additional multiplier of 3 which is applied to the results of the VaR model to convert it into a capital requirement. But using VaR comes at a price. The regulator must ensure that the internal model used to calculate risk is accurate. Otherwise, banks might misrepresent their risk exposure. But, backtesting to check the accuracy of an internal VaR model is dicult. (3) This motivated economists Kupiec and O'Brien (1995b) of the Federal Reserve Board to put forward a new `pre-commitment' approach that proposes the use of a `soft-link'. Such a link is not externally imposed, but arises endogenously, and is induced by the threat of penalties whenever trading losses exceed a level pre-specied by the bank (known as the pre-commitment capital). Specically, under PCA, banks are asked to choose a level of capital to back their trading books for a given period of time (say, a quarter). If the cumulative losses of the trading book exceed the chosen cover at any time during the period, the banks are penalised, possibly by nes. The chosen capital is thus a `precommitment' level, beyond which penalties are imposed. The task of the regulator is to choose an appropriate schedule of penalty to induce a desirable choice of cover for each level of risk. The banks would then position themselves in terms of risk and capital choices for the trading book. The idea is attractive because it does not require the regulator to estimate the level of trading book risk of any particular bank or to approve the rm's model, and promotes more `hands o' regulation. 1.2 Hard links and soft links: a potential trade-o PCA not only circumvents the problems of backtesting, but it gives the banks much greater freedom to choose the portfolios they wish to carry backed by a given level of capital. Since the trading desks of banks are likely to be more adept (historical simulation) models. Parametric approaches are based on the assumption that the distribution of future returns belongs to a given parametric class. The historical simulation approach produces a time series of prots and losses that would have occurred if the portfolio had been held over a specied estimation period. (3) See Kupiec (1995), Jackson et al (1996). 3

7 at estimating risks of various trades, it seems inecient to impose hard links. Indeed, incentive-based regulation such as PCA is inherently more attractive to economists. Though these advantages of PCA have been discussed in the literature, another aspect of this soft-link approach seems to have received little attention. The exibility of a soft-link approach such as PCA derives from the fact that it is not directly prescriptive, but creates incentives through the use of penalties. In more general terms, PCA tries to solve what is known as a `mechanism design' problem. It attempts to specify a mechanism (in this case a set of rules that the banks take into account in choosing portfolio risk and committed capital) that would make it incentive-compatible for the banks to choose the socially desirable risk prole. The success of such a programme depends on how well the regulator anticipates the strategic opportunities that a mechanism might create. In other words, while soft-link approaches are exible and not subject to measurement problems, they create a host of strategic issues. To build a successful soft-link regulatory policy, one must recognise all possible conicts of interest that might arise subsequently, and provide incentives to align them with the objectives of the regulator. The rst step towards building an optimal soft-link policy is to analyse the incentive eects of PCA in a detailed model of the conicts of interest within the bank. This is the objective of the present paper. We have attempted to make precise the content of the above claim that there is a trade-o in switching to PCA from a hard-link approach. We nd that there may be some adverse incentive eects from the adoption of the PCA approach. Optimal incentive-based regulation would presumably attempt to retain the essential features of a soft-link approach such as PCA and combine them with certain properties of a hard-link approach that circumvent certain incentive problems. We discuss some of these issues in the conclusion. But, detailed analysis of more general soft-link approaches is outside the scope of the present paper. This remains an area of our active research interest. In this paper we identify two potential sources of distortion, and analyse their eects under PCA. We describe the problems below. 4

8 1.3 Separation of ownership and control in large banks: the agency problem The Kupiec and O'Brien papers are set in a world where the regulator interacts with the bank. This leaves aside the issue of the eects of the incentive structure within the bank. But, as in most large corporations, an integral feature of large modern banks is the separation of owners from day-to-day decision-making. The ownership is diuse - there are numerous small shareholders who have little impact on most decisions. Even relatively large shareholders would in general have hardly any impact on day-to-day risk taking. It is the incentives of, say, the traders of the bank that determine what specic strategies they might adopt on a particular day. Thus it is important to see to what extent the owners can control their actions. As a device of control, the owners write contracts with managers, and then the managers take most of the trading decisions. Moreover, managers cannot usually be ned (ie paid negative salaries) in the event of losses. (4) So decisions about trading book risk are taken by managers with limited liability, while the owners have to suer the losses (5) in the trading book, and pay the penalty in the case of a breach under PCA. This implies that to study the eectiveness of the incentive structure generated by PCA, it is no longer sucient to consider the bank as a single entity whose actions are directly inuenced by the regulatory incentives. Without explicitly modelling the agency structure and the nature of optimal incentive contracts in the bank, it is dicult to gauge the eect of regulatory policies on large banks. In other words, to evaluate a soft-link regulatory scheme, the appropriate question to ask relates to the eect of the regime on the incentive structure in the bank. An analysis of the response would tell us which regulatory objectives are ltered through, and what aspects need further modication. In this paper, we aim to provide such an analysis. (4) Even when red, most managers are usually able to nd other jobs. (5) Deposit insurance implies a lower limit, but if the trading book is relatively small compared with the banking book, this limit would hardly ever be reached. See section 3.2 for a discussion. In any case, limited liability only adds to the list of trouble spots. 5

9 1.4 Disclosure of breaches: reputational concerns The concern about agency incentives is a general one. To be eective, any regulatory scheme must generate the right incentives throughout the delegated decisionmaking structure within a bank. But a second set of incentive issues applies specically to a feature of PCA. Banks, and especially the large banks, are concerned not just about current earnings, but also about future prospects. So it is important for them to have an impeccable reputation. Maintaining a position of prestige is also often an end in itself. Under PCA, losses exceeding commitment capital are penalised. If a policy of public disclosure of breaches is adopted, it would give rise to concerns by the banks about the eect of breaches on their reputation. Note that this concern is not unrelated to the agency issue discussed above. The manager might not be too concerned about the reputational eects of breaches - but the price of bank shares might indeed have a lot to do with such eects. So in analysing the incentives generated by PCA, reputational issues need to be included. Indeed, our analysis points out the main eects of reputational concerns, and the distortions it might produce. 2 A summary of the results Weinvestigate the above issues in a simple principal-agent framework. We obtain the following results. 2.1 Agency incentives under a hard link approach First, we show that conicts of interest within the bank have no implication for hard-link policies. The regulator sets a capital requirement for each level of risk. At any time the risk cannot exceed the level consistent with the given capital. It is easy to see that this is true irrespective of the incentive structure in the bank. So, while a hard-link regime such as VaR is subject to measurement problems that have been highlighted in the literature, and is economically unattractive in some respects, the presence of a hard link does manage to sort out the potential 6

10 strategic complications. A hard link works because it sets an exogenous legal requirement that cannot be breached. (6) This is not to say that we therefore recommend a hard-link regime. But this property of strict compliance generated by a hard link is worth noting because one might be able to design an incentive-based system that is exible and at the same time builds in some hard links to avoid strategic or reputation-based distortions. 2.2 Agency incentives under PCA The structure of agency would be a concern under any soft-link regime, but the precise eects would dier across dierent soft-link policies. In this paper we analyse the eects of agency on the outcomes generated by PCA. Under PCA, the capital chosen does not constrain the manager's choice of riskiness. In the absence of a priori restrictions on the choice of risk, the outcome depends on the manager's preferences. We show that if managers care only about monetary compensation, the principal (ie the bank owner/shareholders) could design contracts which would generate incentives for the manger to behave consistently with the principal's objectives and in turn the regulator could therefore achieve the right capital levels. But the manager might also be interested in attaining star status by generating large positive returns and so might undertake high-risk strategies (limited managerial liability implies that only the upside matters). We show that in this case tighter controls on the manager can be achieved only at the cost of the principal's own prot. This leads the principal to choose a level of control that is not too tight, resulting in a non-trivial probability of very risky investments and large losses in relation to the amount of capital committed. 2.3 Loss aversion under reputational concerns and distortions in the choice of trading book size Reputational eects of the disclosure of a breach could be quite perverse. For large banks, trading books are small relative to banking books. For such banks, (6) We assume implicitly that the legal system generates incentives enough to ensure compliance irrespective of individual preferences. This is, of course, a standard assumption in many economic studies. 7

11 there is an important asymmetry in the eects of trading book gains and losses above the committed capital. With a relatively small trading book, any gains are on average small relative to the earnings of the bank as a whole. But, the damage to reputation from a breach implies large costs for the entire bank: for example, it could aect funding costs for the whole bank. So gains are in proportion to the trading book size, but losses caused by the damage to reputation are in proportion to the total size of the bank (trading book plus banking book). This asymmetry implies that banks may become `loss averse'. Since a loss is much more important than a gain of the same size, a reduction in the trading book size (whichwould reduce losses and gains symmetrically) increases expected payos. This implies that banks with relatively large banking books would tend to minimize the size of their trading books. Further, such banks would attempt to control the trading book risks tightly - and thus (applying the second result above) sacrice trading book return. Finally, such banks would also tend to over-commit capital. So the banks that are the best candidates (from the social point of view) for carrying larger trading books and undertaking higher trading book risk are precisely the banks that would carry smaller trading books and undertake very low trading book risk. 2.4 Modifying PCA: optimal regulation Finally, we informally discuss possibilities for constructing other soft-link approaches that modify PCA, retaining its exibility and at the same time mitigating the above problems. Reputational distortions are a result of the piecemeal approach. To eliminate such intra-bank allocational distortions induced by reputational concerns, we need to have the same regime, whatever that is, for the whole bank. Second, we argue that a solution to the agency distortions mightbeachieved by manipulating the structure of incentives in a repeated game by hardening the soft link on penalty paths. Formal construction of an incentive-compatible optimal mechanism is beyond the scope of this paper, but it is an area of our active research interest. 8

12 3 The model In this paper we construct a framework to describe the interactions between regulators and banks. The problem is compounded by the fact that economic agents with dierent interests and objectives interact within the same regulated institution. In our model there are three economic agents - the regulator, the principal (bank ownership) and the manager. The preferences of the three are as follows. 1. The regulator seeks to reduce the probability of large losses within an acceptable margin. 2. The principal is risk neutral, and maximizes expected prot. 3. The manager is risk neutral and subject to limited liability. The manager could be one of many types. Some managers care more about the monetary payo while others care more about their careers. The latter type of managers adopt strategies that generate high positive returns for the rm. Since high-risk strategies attach a higher probability to the tails (the lower tail is irrelevant given limited liability), career concerns imply a preference for high-risk strategies. The manager knows his own type, but the principal cannot observe the type of the manager. More formally, the types are described as follows. We denote the type of the manager by a random variable q (remember that the manager's type is a random variable from the principal's point of view). The utility function of a manager of type q puts a weight q on portfolio risk and a weight (1,q) on the monetary payo. Here, q is a random variable with a uniform distribution on [0; 1]. The bank needs to choose both a trading portfolio and a regulatory capital. The riskiness of the trading portfolio is chosen by the manager and is not directly contractable. The capital level is observable and veriable irrespective of who chooses it. Thus in terms of the results here it is immaterial who chooses it. We therefore assume for simplicity that the principal chooses the capital level. The structure of the moves is as follows. The regulator moves rst, and sets the policy regime. The principal moves second, setting the terms of the contract 9

13 with the manager, and setting a capital level. Finally, the manager chooses a portfolio given the incentive structure implicit in the contract. 3.1 The opportunity set The regulator aims to make the banks choose the socially desirable portfolio volatility associated with the capital set aside. Let e V and V 0 denote the change in the value and the current value of the trading book respectively. We assume that the return on the trading book is proportional to the initial investment in the trading book (V 0 ), and has a normal distribution. Thus: ev N(V 0 ;V 0 ): Our model of the diering objectives of the regulator, the bank owners, and the bank manager builds on the following assumption about the opportunity set (the set of possible trading book investment portfolios from which the manager chooses). Assumption 0a In the absence of any regulatory constraint, the preferred portfolio of the bank owner involves high risk. Remark: Indeed, without this assumption there would hardly be any need for regulation. Also, an equilibrium choice of high risk for the whole bank can be derived formally in a model with deposit insurance (thus limiting the downside and creating a \virtual" risk-loving preference). Such issues are tangential to the present analysis, which focuses on internal incentive structures. It would not be dicult to rewrite the model to include deposit insurance, but that would neither change any result nor add to the insights of the paper. So we simply assume, as stated above, that regulation is not vacuous. We also need the opportunity set to satisfy a regularity property which states that portfolios with risk close to zero should earn a return close to the riskless return: Assumption 0b The opportunity set satises the following (right) continuity property - as the riskiness of portfolios decrease to zero from above, the associated returns decrease to the riskless rate. Given the preferences above, assumption 0 (a and b) is all we need to derive our results. Indeed, with normally distributed returns, a sucient condition for 10

14 the assumption 0 is that portfolios with higher variance also yield a higher mean return. Such a relation could either be a basic property of the return structure - deriving from an ecient portfolio frontier, or it might be that given limited liability (through deposit insurance), the conditional mean (conditional on the lower truncation of the distribution of returns) increases with variance. To keep the algebra uncluttered, we make the simplifying assumption that mean and variance are linked by a linear relationship (this is a special case of assumption 0). Assumption 1 =, where is a strictly positive constant. Remark: We emphasize once again that assumption 1 should be treated only as an \as if" device. The assumption of a precise functional form serves to facilitate derivations. All results hold whenever the more general assumption 0 is satised (7). Finally, to keep all solutions nite, we assume that there is a highest possible portfolio variance of 2, which is nite. 3.2 Trading losses and the extent of liability Our analysis of agency problems arising inside the bank is most pertinent for large banks where the ownership is diuse and the separation of ownership and control is a well established fact. For such large banks the size of regulatory capital for the banking book is usually large compared with the trading book capital. Thus a loss arising in the trading book could erode much more capital than initially set aside to back the trading activity. As far as the optimal trading policy is concerned the principal's liability can be considered to be unlimited (8). The regulatory aim, then, is to limit the losses. This is purely a modeling choice - one could translate the large losses into an increase in default probabilities (7) See appendix B for a discussion of this claim. (8) This assumption might seem to be in conict with the statement above that deposit insurance can be modeled easily without aecting the results. However, the point is that the assumption about the extent of liability has nothing to do with the agency-induced tradeos facing the bank. The extent of liability has no implication for the eectiveness of hard-link regimes, while under limited liability PCA is subject to the additional problem of non-payment of nes in case of losses in excess of regulatory capital. Our assumption of unlimited liability simply abstracts away from such problems with PCA to focus only on the agency-induced issues. 11

15 by adding the probability of banking book losses in excess of the total capital. That would not change the analysis here, but add to the notation. 3.3 Cost of regulatory capital The capital level is observable and veriable irrespective of who chooses it. Thus in terms of the results here it is immaterial who chooses it. We therefore assume for simplicity that the principal chooses the capital level. Regulatory capital is usually costly for the bank. This is not surprising. One possible explanation is as follows. Suppose the appropriateness of the assumption that investors are risk neutral derives from the fact that in equilibrium, risk averse individuals can diversify risk. Raising additional capital in order to adopt higher risk trading strategies might create imbalances in the investors' portfolios. As for raising capital from new investors - there is a very large literature in economics exploring capital market imperfections, credit constraints and other distortions facing rms who attempt to raise external capital. For example, adverse selection can create a wedge between the value of internal and external nance (9). Indeed, in the absence of any cost for raising additional capital, optimal regulation is a trivial exercise and does not put any constraints on bank risk-taking. Under any regulatory scheme, the bank could always take arbitrary risks by raising an arbitrarily high capital. In fact, the statement that the standardized approach \overestimates" the level of capital would have no content if equity capital can be raised costlessly. The overestimation is a concern only if there is a more desirable estimate - and a correct estimate is more valuable only when there is a cost of regulatory capital. We denote the cost of regulatory capital by C(K). We assume that C 0 > 0, and C 00 > 0. Thus the cost of capital is an increasing, strictly convex function. (9) For example, suppose the investors do not know the riskiness of the rm they invest in, and therefore a rm must pay a premium to borrow funds. Under certain conditions, the premium per unit increases with the amount borrowed. See, for example,?) for a model of and a discussion of the literature on such distortions in the context of a rm raising capital from a bank. Exactly the same analysis applies when the rm is a bank and it raises capital from outside investors. 12

16 4 Agency incentives under hard-link approaches In what follows, we use the terms hard-link regulation and VaR interchangeably. Under a hard-link regime, the regulator chooses a function that species a level of regulatory capital for each level of risk of the bank portfolio. Here the risk is parameterized by the portfolio standard deviation. Thus the regulator chooses K as a function of. Let K reg () denote this function. We assume that K 0 () > reg 0, K 00 reg () The bank's problem For any given K reg (), we rst analyze the outcome of the internal optimization of the bank. Recall that the principal species the level of capital (the manager does not contribute to the capital of the bank) and writes a contract with the manager, who chooses a portfolio. Limited liability implies that the managerial pay must be non-negative. Note that if the principal chooses any non-negative increasing payo function (increasing with the value of the trading book, e V ), for any choice of, the manager would choose the highest associated. In other words, so long as the payo function is increasing and non-negative, the limited liability constraint is satised, and also the manager's portfolio choice belongs to the ecient frontier. Now, given any regulatory function K reg (), any given any choice of k by the principal, the manager is restricted to choose such that k = K reg ( ): (4.1) For any increasing payo function, the manager will in fact choose =.Thus equation (4.1) is the solution to the problem of the manager under VaR. Note that from equation (4.1), there is a one-to-one correspondence between the principal's choice of k and the manager's choice of. Bychoosing a capital equal to K reg ( ), and any positive increasing payo function, the principal can force the manager to choose =.Thus under VaR, the principal can fully control the manager's choice of by the appropriate choice of k - agency problems do not aect the outcome. The following result summarizes the above discussion. Proposition 1 Under VaR, the information asymmetry between the manager and the principal does not lead to any loss of control by the principal in choosing. The principal has a forcing contract. 13

17 Thus in a VaR regime the principal's choice describes the problem of the bank completely. The principal's gross payo is given by p = e V, C(k), and the principal's objective is to maximize E p = V 0, C(k) subject to the constraint of managerial incentives. Given that =, and the solution to the manager's problem (given by equation (4.1)), we have: E p = V 0, C(K reg ()) (4.2) From proposition 1 above, we know that in solving the principal's problem we can simply solve for the optimal (as any such can be enforced by the right choice of k). From equation (4.2) above, the rst order condition for the optimal for the principal is given by such that V 0 = C 0 (K reg ( ))K 0 reg( ); where the left hand side is the marginal benet from increasing, and the right hand side is the marginal cost of capital multiplied by marginal capital requirement. Thus the right hand side is the cost to the principal arising from a marginal increase in (10). 4.2 Regulatory control under a hard-link regime The objective of the regulator is to specify the maximum probability of making a loss in excess of the capital set aside to back the trading book. If a bank has capital K and the probability that returns fall below,k is set equal to p, the regulatory constraint is given by: Prob(portfolio returns <,K) p, which can be rewritten as Prob(V 0 + e V < V L )=p; (4.1) where the value-at-risk is K VaR = V 0, V L. Now, we know that the manager's optimum is given by equation (4.1). Using equation (4.1), this can be written as: = K V 0 (,1 (1, p), ) : (10) It can be easily checked that given our assumption about the shapes of K reg() and C(), the second order condition is satised. 14

18 See section A.1 in the appendix for details of derivation. Thus the regulatory function is given by K reg () =V 0 (,1 (1, p), ): (4.2) As we show in section A.1 of the appendix, under this rule, a loss of V 0, V L would be exceeded with probability p. Thus this function implements the required regulatory objective. 5 Agency incentives under the pre-commitment approach (PCA) Under the PCA, the regulator species a ne as a function of any loss over and above the committed capital. Suppose the committed capital is given by K c. Then the ne function is denoted by f(,k c, e V ). For simplicity,we will assume that the ne is a linear function of the breach: f(,k c, e V )=(,d(kc + e V ) if e V <,K c 0 otherwise. This assumption is purely simplifying, and all the results would hold for any other increasing ne function. In this section we show that the agency problem inside the bank is no longer irrelevant to the choice of in situations where the manager cares not just about the monetary payo but also about own reputation (i:e: the manager puts positive weight on the variance for the trading-book portfolio - higher variance strategies increases the chance of higher returns). First, note that the agency problem would not arise here if the standard deviation is observable and contractable - in that case the principal could specify the to be chosen by the manager. Then the problem would be exactly the same as under a hard-link approach and thus the resulting choice of riskiness would be the same. Now suppose is not observable. This introduces an agency problem. Recall from the specication of the model that there are managers of various types. The utility function of a manager of type q puts a weight q 2 [0; 1] on the standard-deviation of the bank portfolio and a weight (1, q) on the monetary payo. Let g m ( e V ) denote the payo function of the manager. The utility function 15

19 of the manager is given by U m =(1, q) g m ( e V )+q (5.1) Recall that q is a random variable with a uniform distribution on [0; 1]. Further, limited liability implies that g m () 0. First, we consider the special case where the manager only cares about his monetary payo. 5.1 Agency incentives under PCA: the special case of q = 0 If q = 0, the manager only cares about monetary payo. We show below that in this case, the principal can design a contract to enforce any. Thus asymmetry of information (choice of by the manager not observable) makes no dierence to the control by the principal, and thus, in turn, to regulatory control. Consider a payo function ^g m ( e V ) given by the following: ^g m ( e V )= 8 >< >: W for V e 1 2r 2 (1 + r V e )W for, 1 3 r < V e < 1 2r 0 otherwise where W is the top salary level. In the above function the scale factor and the origin have been chosen for convenience. Note that in keeping with limited liability of the manager, the payo function is always greater than or equal to zero. Figure 1 shows a picture of this. The parameter r controls the induced level of prudence of the manager when selecting the optimal investment strategy. Section A.2 in the appendix shows that the expected payment of the manager is a function of r and only through the term r. Let r. Then, E^g m ( e V )=t(; ): (5.2) Dierentiating with respect to and solving the rst order condition for a maximum, we get such that t 1 ( ;) = 0, where the subscript 1 denotes derivative with respect to the rst argument. Thus is some constant T 0, and = T 0 r : 16

20 Pay 6 W 2 3 W, 1 r 0 1 2r - V Figure 1: The manager's payo function 17

21 Thus by varying r, anychoice of can be enforced. If the principal can control the manager fully, it is easy to see that a soft-link approach generates no further incentive problems, and the regulatory problem once again becomes (virtually) a problem of interaction between the regulator and the bank as a whole. The reason, briey, is as follows. The expected payo of the principal (section A.3 derives a formal expression for the payo) depends on two factors. First, the higher the committed capital, the higher the cost of capital, but lower the expected ne payment. Second, the expected ne payment depends also the managerial choice of (which in turn depends on r). An increase in implies a higher expected ne, but it also implies a higher expected return. These tradeos decide the optimal choice of K c and r (which controls ). The principal maximizes his expected payo with respect to K c and, and enforces the optimum by choosing r such that T 0 =r =. Since the principal is aected by the ne, and a greater ne makes the principal more conservative in terms of his choice of, the regulator can enforce any desired level of by choosing the appropriate ne function. In this case, any outcome that can be obtained under a hard-link regime can also be achieved under PCA. In fact, all the usual advantages of PCA apply - and there is no downside. 5.2 Agency incentives under PCA: the general case When q>0, the manager cares not just about monetary payo, but also about high-risk strategies in order to generate high positive returns for the bank. Consider any payo function g m ( e V ) 0, where the inequality reects the limited liability constraint. The manager's utility function is now given by equation (5.1). The rst order condition for expected utility maximization m ( e V =,! q V 0 : (5.3) 1, q As q! 1, the right hand side approaches,1. Now, if the payo function is such that the left hand side is bounded below, there exists a q 2 (0; 1) such that for q>q, the rst order condition cannot be satised in the interior (11). - and thus the solution would be at. Thus all types in (q ; 1] would adopt. This (11) If L denotes the lower bound of the left hand side, q is given by L =(L, V 0 ). Note that L < 0 and V 0 > 0, so that q 2 (0; 1). 18

22 implies that in order to keep control of the manager, the principal would like to choose a payo function such that it could V e ) unboundedly low at > 0. Then all types would choose some 2 (0; ). Note that control is parametrized by q. Tighter control implies a higher q. An optimal payo function would therefore attempt to maximize control. However, as the following result shows, in attempting to tighten control, the principal faces a tradeo. Proposition 2 For any optimal payo function g m m ( e V = M 0 where M 0 < 0 and nite and independent of. Proof: See appendix A.4. 2 This has important implications. First, as we noted above, to tighten control (i:e: to raise q ), we need to lower the left hand side of equation (5.3). But the proposition tells us that the way to do that is by implementing a lower. In particular, the term is unbounded below only as approaches zero. This proves the following corollary. Corollary 1 As we approach full control over all types (ie q! 1),! 0. This, in turn, implies! 0. Remark: This key result shows the trade-o faced by the principal. On the one hand, increasing control lowers the weight on. On the other hand, as the control increases, and q goes to 1, goes to zero for all q q.thus to make the agency distortion very small, the principal must choose a payo function that implies a very high control. But at such a high degree of control, the average return is very small. 6 Agency distortion under PCA Under a hard-link regime, the regulator can enforce optimum loss probabilities irrespective of the managerial utility function. But this is not the case under 19

23 PCA. The reason is as follows. Under standard assumptions, the principal's choice problem is well dened, and has interior solutions for K c and. This implies that the principal chooses to set q < 1. Thus there is a non-trivial probability (given by (1, q )) that a very high-risk portfolio (with =) is chosen by the manager. This proves the following result. Corollary 2 Under the pre-commitment approach, the managerial choice of equals with probability =(1, q (r)), where is bounded away from zero. Under a hard-link regime, the principal chooses a given the regulatory function K(). We showed that the regulator can enforce any as the principal's optimum, and the principal in turn can control the manager fully - thus the principal's optimum is implemented as the nal outcome. Under PCA, the rst step is still true. The regulator can enforce any as the principal's optimum through the use of punishments in the event of a breach of commitment capital. But the second step is no longer true. To implement any >0, the principal must necessarily allow q < 1. Thus there remains a probability of(1, q ) with which avery high-risk portfolio is chosen. Thus when the choice of by the manager is not contractable, and the manager's utility depends not just on managerial salary but also on career concerns or preference for attaining star-status, the expected losses increase under the pre-commitment approach (compared to a hard-link approach). 7 Public disclosure of breach: reputational distortion under PCA We now proceed to the second source of distortion arising under PCA. When the breach (under PCA) is publicly announced, this can have a serious impact on the reputation of the bank, which leads to lowering of its future profitability (12). In this section we discuss the impact of such reputational concerns on banks. (12) Even if the breach is not disclosed publicly, once an institution has been ned signicantly, the cost that the penalty generates has to be reported in the balance sheet. This implies that large breaches would probably be uncovered (even though with some delay). 20

24 The rst point to note is that when discussing reputational concerns, we must include into our calculation the value of the entire bank - not just the trading book. Even though the reputational concerns arise from a potential breach with respect to the trading book capital, the eect of a poor reputation aects also the banking book. With any breach, whenever there is less than full deposit insurance, the cost of both debt and equity funding rises. This reduces the return of the trading portfolio, but even more importantly, makes the value of the banking book sensitive to the variance of the trading book. A higher variance induces a higher probability of a breach, and consequent loss of reputation. The above considerations create an asymmetry between gains and losses. Suppose a bank has a large banking book relative to the trading book. While a prot on the trading book would not be very important for the overall return of the bank, a loss on the trading book in excess of the precommitted capital would have avery large reputation eect on the entire value of the bank. Thus a small trading book can jeopardize the large banking book. This makes the bank lossaverse with respect to the trading book and leads the bank to marginalize its trading book even further, and possibly also to overcommit trading book capital. 7.1 An illustration Let us give an illustration of the kind of distortions that reputational concerns introduce in a very simplied setting. Let the current value of the bank be denoted by V 0 B. Then V 0 B = V 0 bb + V 0 tb; where the subscript \bb" denotes banking book and the subscript \tb" denotes trading book. Suppose V 0 bb = V 0 tb; >0: Now, normalize V 0 0 tb to unity. Then V =, and V 0 =1+. bb B Let e V bb and e V tb denote the returns on the banking and trading books, respectively. Assume that all returns are proportional to the current values. Suppose these are distributed as follows: ev tb N(; ); 21

25 and ev bb N(; ^): Suppose the bank commits a capital of K c = kv 0 tb = k. Suppose also that the cost of capital is a linear function C(k) =ck, c>0. Whenever the trading book risk >0, a breach occurs with probability Prob( e V tb <,K c )=,k, where is the standard normal distribution function. Note that here the probability of breach is independent of. Let C R denote the reputational cost. As we argued above, the cost of reputation is proportional to V 0 B =(1+). Assume that C R is linear in V 0 B. We have: where R is a positive constant. C R = RV 0 B = R(1 + ); If the trading book risk >0, a breach might occur. The bank earns a return of + whether a breach occurs or not, and loses R(1 + ) if a breach occurs. Thus the expected return of the bank is given by E e V B j>0 = +,,k,!! (1 + )R, ck (7.1) On the other hand, if the bank undertakes no trading book risk, i:e: =0, there is no chance of a breach and the issue of reputational cost does not arise. Then the expected return is given by E e V B j =0 = E e V bb, ck =, ck (7.2) Let us write the breach probability as simply. From equations (7.1) and (7.2), E e V B j>0 E e V B j =0 if and only if This implies the following result. =, R R : Result: Banks with a relatively large banking book ( > ),would prefer to reduce their trading book risk to zero. Only banks with relatively small banking books would undertake risky trading strategies. 22

26 Remark: These extreme conclusions are, of course, a result of this particularly simple set-up - but the insight is robust. The asymmetry in losses and gains introduced by reputational concerns induces a bank with a relatively small trading book to reduce its importance further by reducing trading book risk. On the other hand, banks with relatively small banking books have little to lose in terms of reputation, and these banks would therefore undertake higher trading book risk. Banks with a large banking book relative to the trading book are the best candidates (from the point of view of social optimality) to bear greater trading book risks. Yet these are precisely the banks that would reduce their trading book risk, and possibly commit to very large trading book capital. On the other hand, banks with small banking books presumably have limited liability - they cannot be ned very high amounts once they have suered a large loss on the trading book. Such banks would therefore care more about trading book prots than losses - they would tend to increase the size of the trading book risk. Thus a perverse pattern emerges. Large banks with a large banking book relative to the trading book would be very loss averse, and would either make the trading book risk insignicantly small and possibly overcommit capital. In either case, since social optimality requires precisely these banks to take higher risk, reputational concerns generate ineciency. 8 Conclusion: modifying PCA We conclude by discussing possible remedies. First, let us briey review the results. Under a hard-link regime (with the exogenous link K reg () mapping to capital requirement), the principal's optimum choice of can be enforced by choosing any payo function increasing in V and a capital cover K such that = K,1 reg(k). Thus bychoosing the rightschedule of required capital, the regulatory choice of loss probabilities can be implemented. Under PCA, if the manager's utility only depends on monetary payo, the principal retains full control of the manager, and his optimum choice (K ; ) can be enforced by choosing K = K and r such that (r )=. The choice of K and depends on the ne schedule. Thus by choosing the right schedule of ne, the regulatory choice of loss probabilities can be implemented. However, if the manager also has career objectives - so that the manager places some weight q on a higher trading book standard-deviation (where q is 23

27 unknown to the principal, who only knows the distribution of q), the agency problem generates higher expected losses. This is because, as we have shown, whenever the principal attempts to implementa>0, there is a certain q such that for all types q>q there is a jump in the variance undertaken. Thus the agency problem creates distortions in trading book risk. Further, reputational concerns introduce distortions in the choice of trading book size relative to the banking book across the banking sector. Modifying PCA to rectify the distortions The reputational distortion arises because the regulation treats the trading book separately from the banking book. Any piecemeal regulation that attempts to punish trading losses would generate an asymmetry in the bank's evaluation of trading gains and losses, and make the bank loss averse. Of course, the obvious way to correct for this distortion is to regulate the whole bank under the same regulatory scheme. Correcting for agency distortions is more complicated. In general, this is a problem of designing a mechanism to implement a certain objective given that various interacting agents have conicting preferences (13). Such a general approach could be very fruitful in this context, and while this is one of our research areas, an analysis along this line is beyond the scope of the paper. However, there is another possible route - since the interaction between the regulator and the banks takes place repeatedly over time, we need not focus simply on static regulation. The key problem here is that on the one hand, maintaining exibility makes it necessary to allow the banks to choose their own riskiness, and on the other hand, such exibility might result in loss of control by the principal over the manager. A hard-link is inexible, but it allows full control. A loss of control occurs when managers of dierenttypes have dierent preferences for portfolio risk. In view of this, we might attempt to retain the exibility and yet harden the soft-links under PCA in the following manner. Consider the following scheme for any given bank: (13) For a lucid discussion of the central issues in the implementation literature, see the survey by Moore (1992). 24